Solving for Bending Stress and Deflection in a Cantilever Plate

[FEAnalyst]

TL;DR

How to calculate stress and deflection of a cantilever plate analytically ?

Hi,

books such as „Roark’s Formulas for Stress and Strain” or Timoshenko’s „Theory of plates and shells” provide formulas for maximum bending stress and deflection for many cases of rectangular plates. However, I can’t find a simple case of a cantilever plate (one edge clamped, all other edges free) subjected to UDL anywhere. The closest example is a plate with one edge clamped and all other edges simply supported (from Timoshenko’s book) but it’s not the same. There must be a way to solve this problem of a shelf-like plate analytically. Do you know where I can find appropriate formulas ?

Thanks in advance for your help

 

[Dr.D]

UDL??

 

[Chestermiller]

This is a pretty tricky problem because of the lateral constraint of the plate at the clamping. If it were allowed to slide laterally, that would make it much easier. I'm sure that, somewhere in the literature, this problem (more complex version) has been solved, but I have never researched it.

 

[FEAnalyst]

UDL??

What I mean is uniformly distributed load.

This is a pretty tricky problem because of the lateral constraint of the plate at the clamping. If it were allowed to slide laterally, that would make it much easier. I'm sure that, somewhere in the literature, this problem (more complex version) has been solved, but I have never researched it.

Thanks. I noticed that this problem is much more complex than it seems. I know that such plate can be treated as a beam but it's not very accurate approach. Maybe it's possible to solve this case using trigonometric series or other complex method but I'm not sure how to do it.

 

[FEAnalyst]

I've found a solution in a Polish book "Collection of tasks on strength of materials" by Banasiak, Grossman and Trombski. This book features a derivation and the final equations are: $$\displaystyle{ \sigma_{x \ max}=\frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{y \ max}=\nu \cdot \frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{vM \ max}=\frac{3qa^{2}}{h^{2}} \cdot \sqrt{1+ \nu^{2}- \nu}}$$ $$\displaystyle{ w_{max}=\frac{3 (1- \nu^{2})qa^{4}}{2 E h^{3}}}$$
where: $q$ - magnitude of uniformly distributed load, $a$ - length of the plate (from fixed to free end), $h$ - thickness of the plate.
The results are in very good agreement with FEA (much better than when the plate is treated as a beam).

 

[Chestermiller]

I've found a solution in a Polish book "Collection of tasks on strength of materials" by Banasiak, Grossman and Trombski. This book features a derivation and the final equations are: $$\displaystyle{ \sigma_{x \ max}=\frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{y \ max}=\nu \cdot \frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{vM \ max}=\frac{3qa^{2}}{h^{2}} \cdot \sqrt{1+ \nu^{2}- \nu}}$$ $$\displaystyle{ w_{max}=\frac{3 (1- \nu^{2})qa^{4}}{2 E h^{3}}}$$
where: $q$ - magnitude of uniformly distributed load, $a$ - length of the plate (from fixed to free end), $h$ - thickness of the plate.
The results are in very good agreement with FEA (much better than when the plate is treated as a beam).

What are the edge boundary conditions for this solution?

 

[FEAnalyst]

What are the edge boundary conditions for this solution?

One edge of the plate is fixed, the remaining ones are free:

 

[Chestermiller]

One edge of the plate is fixed, the remaining ones are free:
View attachment 291921

On the constrained edge, is it constrained transversely so that the strains in the y direction are zero at this boundary?

 

[FEAnalyst]

On the constrained edge, is it constrained transversely so that the strains in the y direction are zero at this boundary?

The boundary conditions are given in the book as follows: $$w_{x=a}=0$$ $$\varphi_{x=a}=\left( \frac{dw}{dx} \right)_{x=a}=0$$ $$t_{x=0}=-D \left( \frac{d^{3}w}{dx^{3}} \right)_{x=0}=0$$ $$\left( m_{y} \right)_{x=0}=-D \left( \frac{d^{2}w}{dx^{2}} \right)_{x=0}=0$$
where: $t$ - shear force, $m$ - bending moment.